The detection of anomalous events in noisy systems remains a fundamental challenge across physics, information theory, and complex systems. Conventional approaches typically rely on local statistical deviations, which often fail in non-stationary environments or in processes with long-term memory. In this work, we introduce luck as an informational observable that explicitly incorporates historical context into anomaly detection. Starting from an original convolution-based formulation involving the Dirac delta, luck is defined as the ratio between the local probability of an event and the cumulative probability accumulated by the system up to that point. Unlike probability, luck is not bounded by unity and is undefined in the absence of history, reflecting the principle that anomalies require memory. By normalizing luck, we obtain a distribution that characterizes how unexpected information is allocated across events. The associated entropy of luck provides a global measure of the informational structure of a process, distinguishing between regular, quasi-deterministic dynamics and intermittent regimes dominated by rare, history-breaking events. Through numerical experiments with strong noise and synthetic examples inspired by real-world systems, we demonstrate that luck-based analysis robustly suppresses background fluctuations while selectively amplifying genuine anomalies. Comparisons with classical local detectors show that the proposed framework operates without arbitrary thresholds and remains effective in highly noisy and non-stationary conditions. These results establish luck as a principled, computationally efficient, and broadly applicable informational observable – distinct from probability – that quantifies historical mismatch rather than frequency, making it suitable for anomaly detection in systems with memory.
Carlos Riveros Berger (Thu,) studied this question.